Published on June 12th, 2013 | by Emily Corbett0
Puzzle Challenge 5
This months puzzles have been provided by Michael Evans from AMSI
The winner of this months puzzle of the month will receive a signed copy of a Keith Devlin book and a $50 voucher, to enter email full working to all puzzles to firstname.lastname@example.org by 31 August. Entries will be judged on number of correct answers, originality and presentation of ideas.
1. There is a fixed volume of water available. There are n jugs, each big enough to hold all of the water. Initially all of the jugs contain the same amount of water. Water can be poured from any jug A to any other jug B so that the amount poured from jug A to jug B is the same as the amount of water in jug B. For which values of n is it possible to collect all the water in one jug?
2. Show that for every sufficiently large integer n, it is possible to split the integers 1, 2,…, n into two disjoint subsets such that the sum of the elements in one set equals the product of the elements in the other.
3. There are 2N people at a party. Each knows at least N others. Prove that one can always choose four people and place them at a round table so that each person knows both neighbours.
4. In a regular octagon each side is coloured blue or yellow. From such a colouring a new colouring is obtained in one step as follows: If two neighbours of a side have a different color the new colour of that side will be blue, otherwise it will be yellow (the colours are modified simultaneously).
a. Show that after finitely many moves all sides will be coloured yellow.
b. What is the maximum number of moves that may be needed to achieve this state?